As can be seen in Fig. REF , we find that for a crosstalk error-rate of \(p_c=10^{-6}\) , or below, a pseudo-threshold can be obtained when the entangling MS-gate error lies below \(p_{\rm MS}\lesssim 2\cdot 10^{-3}\) , a value that is at reach for current technology. Let us also remark that, contrary to idealised noise models with a single error parameter, the regime of quantum logic advantage is contained within a range of physical entangling-gate error rates. If the MS-gate error is reduced well below the levels of the other noise sources, it will no longer be possible to demonstrate quantum logic advantage by comparing the logical error rate with the MS-gate error rate, as the other sources of noise will be the leading ones. This leads to a plateau-like levelling out of the logical error rate, similar to the behaviour observed in Ref. [1]}. This clarifies the inherent complexity of having a single well-defined threshold for realistic multi-parameter noise models, which justifies the search for other metrics to quantify the benefit of QEC [2]}, [3]}. Although the required MS-gate error level is realistic for current state-of-the-art architectures, reducing crosstalk error rates to values on the order of \(10^{-5}\) -\(10^{-6}\)  might be challenging using current optical addressing techniques. This difficulty motivates using refocussing techniques, as an additional circuit level technique to further suppress residual crosstalk. As we will show in the following sections, the application of a refocussing scheme will allow for a reduction of crosstalk levels by several orders of magnitude, leading the scheme back to a beneficial QEC regime. Further reduction of crosstalk to values below \(c\simeq 10^{-7}\)  does not improve the results, as its effects become negligible with respect to the other error sources present in the simulations. Although crosstalk errors still have the capability to break FT in the Steane code, as a single crosstalk fault event can lead to a non-correctable error, it stops being the limiting factor for the performance of QEC in the experimentally relevant regime.
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